Question

A concentrate sphere of radius R has a cavity of radius r which is packed with saw-dust. The specific gravity of concrete and sawdust are $ 2.4 $ and $ 3.0 $ respectively. The sphere floats with its entire volume submerged under water. Calculate the ratio of mass of concrete and the mass of saw- dust
(A) 1
(B) 2
(C) 3
(D) 4

Answer 1

Hint : To solve this question, we have to know about the mass. It is one of the key amounts in Physics and the most essential property of issue. We can characterize mass as the proportion of the measure of issue in a body. The SI unit of mass is Kilogram (kg).

Complete Step By Step Answer:
Let us consider the specific gravities of the concrete and saw dust are $ {\rho _1} $ and $ {\rho _2} $ respectively.
According to the question,
$ {\rho _2} = 0.3 \\
  {\rho _1} = 2.4 \\ $
According to the floatation, weight of the whole sphere is equal to upthrust on the sphere.
 $ \dfrac{{4\pi }}{3}({R^3} - {r^3}){\rho _1}g + \dfrac{{4\pi }}{3}{r^3}{\rho _2}g = \dfrac{{4\pi }}{3}{R^3} \times 1 \times g $
Or, $ ({R^3} - {r^3}){\rho _1} + {r^3}{\rho _2} = {R^3} $
Or, $ {R^3}{\rho _1} - {r^3}{\rho _1} + {r^3}{\rho _2} = {R^3} $
Or, $ {R^3}({\rho _1} - 1) = {r^3}({\rho _1} - {\rho _2}) $
Or, $ \dfrac{{{R^3}}}{{{r^3}}} = \dfrac{{{\rho _1} - {\rho _2}}}{{{\rho _1} - 1}} $
Or, $ \dfrac{{{R^3} - {r^3}}}{{{r^3}}} = \dfrac{{{\rho _1} - {\rho _2} - {\rho _1} + 1}}{
  {\rho _1} - 1 $
Or, $ \dfrac{{({R^3} - {r^3}){\rho _1}}}{{{r^3}{\rho _2}}} = (\dfrac{{1 - {\rho _2}}}{{{\rho _1} - 1}})\dfrac{{{\rho _1}}}{{{\rho _2}}} $
Mass of concrete / mass of saw dust =
$ \dfrac{{1 - 0.3}}{{2.4 - 1}} \times \dfrac{{2.4}}{{0.3}} \\
   = 4 \\ $
So, the ratio of mass of concrete and the mass of saw dust is $ 4 $
So, the right option is option number $ 4 $ .

Note :
We know that the mass of a body doesn't change whenever. Just for certain limit situations when a gigantic measure of energy is given or taken from a body. We also know that Explicit Gravity or relative gravity is a dimensionless amount that is characterized as the proportion of the thickness of a substance to the thickness of the water at a predefined temperature.